Inverse Initial Boundary Value Problem for a Non-linear Hyperbolic Partial Differential Equation

TL;DR

This paper studies an inverse problem for a non-linear wave equation, showing that potential and quadratic coefficients can be uniquely identified from boundary measurements over finite time, extending linear inverse problem techniques.

Contribution

It introduces a method to determine the potential and quadratic coefficients in a non-linear wave equation from boundary data, based on linearization at the trivial solution.

Findings

Unique determination of potential and quadratic coefficients from boundary measurements.

Linearization approach at the trivial solution simplifies the non-linear inverse problem.

Boundary measurements over finite time are sufficient for coefficient recovery.

Abstract

In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension $n\geq 2$. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear wave equation with a time independent potential. For any small solution $u=u(t,x)$ of this non-linear equation, it is the perturbation of linear wave equation with time-independent potential perturbed by a divergence with respect to $(t,x)$ of a vector whose components are quadratics with respect to $\nabla_{t,x} u(t,x)$. By ignoring the terms with smallness $O(|\nabla_{t,x} u(t,x)|^3)$, we will show that we can uniquely determine the potential and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval and the final overdetermination at $t=T$. In other words, the measurement is given by the so-called the input-output map (see (1.5)).